For a finite dimensional C * -algebra A and any C * -algebra B, we determine a constant of equivalence of operator space projective norm · ∧ and the Banach space projective norm · γ on A ⊗ B. We also discuss the * -Banach algebra A ⊗B.Mathematics Subject Classification (1991):46L05, 46C10, 47035

Let A and B be C * -algebras. We prove the slice map conjecture for ideals in the operator space projective tensor product A ⊗ B. As an application, a characterization of the prime ideals in the Banach * -algebra A ⊗ B is obtained. In addition, we study the primitive ideals, modular ideals and the maximal modular ideals of A ⊗ B. We also show that the Banach * -algebra A ⊗ B possesses the Wiener property and that, for a subhomogeneous C * -algebra A, the Banach * -algebra A ⊗ B is symmetric.2010 Mathematics subject classification: primary 46L06; secondary 46L07, 47L25.

For C * -algebras A and B, the identity map from A ⊗B into A⊗ λ B is shown to be injective. Next, we deduce that the center of the completion of the tensor product A ⊗ B of two C * -algebras A and B with centers Z (A) and Z (B) under operator space projective norm is equal to Z (A) ⊗Z (B). A characterization of isometric automorphisms of A ⊗B and A⊗ h B is also obtained.

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We prove an analogue of Hardy's Theorem for Fourier transform pairs in ℝ for arbitrary simply connected nilpotent Lie groups, thus extending earlier work on ℝn and the Heisenberg groups ℍn.